Generalized Donaldson-Thomas theory over fields K $\ne$ C

Abstract

Generalized Donaldson-Thomas invariants defined by Joyce and Song arXiv:0810.5645 are rational numbers which `count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold $X$, where $\tau$ denotes Gieseker stability for some ample line bundle on $X$. These invariants are defined for all classes $\alpha$, and are equal to the classical Donaldson-Thomas invariant defined by Thomas arXiv:math/9806111 when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. Joyce and Song use gauge theory and transcendental complex analytic methods, so that their theory of generalized Donaldson-Thomas invariants is valid only in the complex case. This paper will propose a new algebraic method extending the theory to algebraically closed fields K of characteristic zero, and partly to triangulated categories and for non necessarily compact Calabi-Yau 3-folds under some hypothesis.

arXiv commentpreliminary version to appear in my PhD thesis. arXiv admin note: substantial text overlap with arXiv:0810.5645 by other authors

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Cited by in corpus (9)

• Derived Algebraic Geometry and Deformation Quantization
• Hall algebras in the derived category and higher rank DT invariants
• Torus localization and wall crossing for cosection localized virtual cycles
• Gopakumar-Vafa invariants and wall-crossing
• Gopakumar-Vafa invariants via vanishing cycles
• On motivic Joyce-Song formula for the Behrend function identities
• The Thom-Sebastiani theorem for the Euler characteristic of cyclic L-infinity algebras
• Donaldson-Thomas invariants of Calabi-Yau orbifolds under flops
• Note on the motivic DT/PT correspondence and the motivic Flop formula